The election took place Sunday despite a recent wave of deadly violence in the east and threats by pro-Russia separatists to prevent citizens from casting their ballots.

Voters were picking a successor to ousted pro-Moscow President Viktor Yanukovych in a country torn apart by Russia's takeover of Ukraine's Crimea peninsula and bloody conflict with pro-Russia factions.

The unrest has centered in the Donetsk and Luhansk regions, where separatists have claimed independence following a disputed referendum earlier this month -- and many there did not get to cast ballots Sunday.

As of 3 p.m. (8 a.m. ET), some 528 polling sites out of 2,430 were open in the Donetsk region, the regional administration said. Local officials said there was 11.8% turnout at these polling stations.

Ukraine votes for new president

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Outside the country's restive east, voting was progressing more normally.

The Central Election Commission put voter turnout at nearly 38% as of 3 p.m. local time, not including the Donetsk and Luhansk regions, Ukraine's official Ukrinform news agency reported.

In the city of Donetsk, the regional capital where pro-Russia militias are concentrated, there were no open polling stations, local officials said earlier.

A CNN team driving through the city Sunday morning was not able to see a single polling station in operation. However, there were signs some voters were trying to go to polling stations in areas west and south of the city.

Besides the presidential race, candidates are also running in municipal elections in some cities. The Organization for Security and Co-operation in Europe deployed 900 observers for the election -- the largest such mission in its history.

Oleksandr Turchynov, Ukraine's acting President, called the polls open and transparent.

"The voting was free, without artificial restrictions and administrative pressure," Turchynov said in a statement.

U.S. President Barack Obama congratulated Ukrainians for casting their ballots Sunday and criticized Russian-backed separatists, whom he accused of trying to block voting.

"Despite provocations and violence, millions of Ukrainians went to the polls throughout the country, and even in parts of eastern Ukraine, where Russian-backed separatist groups sought to disenfranchise entire regions, some courageous Ukrainians still were able to cast their ballots," he said in a written statement.

"We commend the resolve of all those who participated, as well as the efforts of the Ukrainian government to conduct these elections in the face of those threats."

Intimidation in eastern Ukraine appears widespread

Increasing violence in the east has led the authorities in Kiev to accuse Russia, which they say is backing the armed separatists, of seeking to disrupt the vote. Russia denies having direct influence over the militants, and Russian President Vladimir Putin has said he will respect the Ukrainians' choice.

A large separatist rally was held in a central Donetsk city square around lunchtime. The protesters, who chanted pro-Russia slogans as they were addressed by separatist leaders, were joined by a substantial number of militants on trucks, some firing guns into the air.

On the back of some of the trucks were armed men who appeared to be Chechen. Two told a CNN team they were from the Chechen capital, Grozny, and one indicated that he was formerly a policemen in Chechnya and was in Donetsk to serve the Russian Federation.

The men, who as Chechens are Russian citizens, said they were there as "volunteers." But if their accounts were true, their presence in Donetsk would appear to indicate some kind of acquiescence by the Russian government at the least.

Residents of Ukraine's southeastern city of Mariupol saw new billboards on the streets Sunday urging them not to cast their ballots. The billboards were not at those locations the night before, residents said.

Also in Mariupol, people talked on social media about being asked by local Russia supporters to boycott the election. The city is one of several where deadly clashes have erupted in recent weeks.

The self-declared mayor of rebel stronghold Slovyansk, Vyacheslav Ponomaryov, has said that anyone who tries to vote there will be arrested.

Also, an Italian journalist was killed Saturday near the flashpoint town, the Italian Foreign Ministry announced Sunday. The man, named as Andrea Rocchelli, was killed along with a Russian citizen, the ministry said. Reports suggested there had been mortar fire in the Slovyansk area.